Fundamental Theorem of Arithmetic
and the Infinitude of Primes.
• The Number Theory of Euclid: Burton - 4.3, Journey - 3
– Introductory Definitions
– Divisibility and the Greatest Common Divisor
– Euclidian Algorithm
• The Infinitude of Primes: Burton - 4.3, Journey - 3
• The Fundamental Theorem of Arithmetic: Burton - 4.3
• A Few Comments on Primes
– Sieve of Eratosthenes: Burton - 4.4
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– Fermat, Euler, and primes of the form 22 + 1: Journey - 10
– Theorems of Euler and Fermat: Journey - 10
– The Prime Number theorem: ???
– The Twin Prime Conjecture: ???
– The Goldbach Conjecture: ???
– Fermat’s Last Theorem: Burton - 10.2
• Assignments: Show All Work
– Sec. 4.3 - 2,6,10,14,15,16 cd
– Carl Friedrich Gauss is a pivotal figure in the history of Number Theory. Write a 1 to 2
page essay about Gauss using the same guidelines as the unit 1 essay.
• Test Materials:
– Where, when, & who?:
Create a “cheat sheet” with the names, dates, locations, and one or two significant facts
about each of the following individuals.
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Euclid
Eratosthenes
Euler
Fermat
Gauss
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Andrew Wiles
Diophantus
Mersene
Sophie Germain
Others
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Composite (Euclid’s Def.)
Odd (Euclid’s Def.)
Even (Euclid’s Def.)
Perfect Numbers
Others
– What does it mean?
Define each of the following terms.
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Divisor
Greatest Common Divisor
Least Common Multiple
Unit (Euclid’s Def.)
Prime (Euclid’s Def.)
– Practice Problems: Sec. 4.3 - 1,2,4,10,11,12,15,16, Other
– Proofs:
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Prove that there are infinitely many primes.
Outline the proof of the Fundamental Theorem of Arithmetic.
Outline the proof of Euler’s Theorem.
State the Prime Number Theorem.
State Fermat’s Last Theorem.